A large isolated metal sphere of radius r carries a fixed charge. A small charge is placed at a distance d from its surface. It experiences a force which is

Explanation

The correct option is:(d) Inversely proportional to (r + d)²Here's why each option is correct or incorrect:(a) Independent of r and d: This is incorrect because the force experienced by the small charge depends on both the charge of the sphere and the distance between the charge and the sphere. As the distance increases or the sphere's charge changes, the force will also change.(b) Proportional to r² + d²: This is incorrect because the force is not directly proportional to the sum of the squares of the radius and the distance. While both r and d play a role, their relationship with the force is more complex.(c) Proportional to r²: This is incorrect because the force depends not only on the sphere's charge (which is related to its size) but also on the distance between the small charge and the sphere. As the distance increases, the force weakens even if the sphere's size remains constant.(d) Inversely proportional to (r + d)²: This is correct. Here's the explanation:Gauss's Law: We can apply Gauss's Law to the situation. Since the metal sphere is a conductor, the excess charge will reside on its surface, creating a uniform electric field inside and outside the sphere.Electric Field Due to a Sphere: The electric field due to a uniformly charged sphere at a distance d from its center is given by:E = k * Q / [(r + d)²]where:E is the electric field strength (N/C)k is the Coulomb constant (8.99 × 10^9 N⋅m²/C²)Q is the total charge on the sphere (C)r is the radius of the sphere (m)d is the distance from the center of the sphere (m)Force on the Charge: The small charge (q) placed at this distance experiences a force due to the electric field (E):F = q * ESubstituting the expression for E from the previous equation:F = q * [k * Q / [(r + d)²]]Simplifying, we get:F = k * Q * q / [(r + d)²]This shows that the force (F) is indeed inversely proportional to the square of the sum of the radius (r) and the distance (d), making option (d) the correct answer.incorrect:(a) Independent of r and d: This is incorrect because the force experienced by the small charge depends on both the charge of the sphere and the distance between the charge and the sphere. As the distance increases or the sphere's charge changes, the force will also change.(b) Proportional to r² + d²: This is incorrect because the force is not directly proportional to the sum of the squares of the radius and the distance. While both r and d play a role, their relationship with the force is more complex.(c) Proportional to r²: This is incorrect because the force depends not only on the sphere's charge (which is related to its size) but also on the distance between the small charge and the sphere. As the distance increases, the force weakens even if the sphere's size remains constant.